L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s + 2·13-s + 15-s + 16-s + 6·17-s − 18-s − 4·19-s + 20-s − 23-s − 24-s + 25-s − 2·26-s + 27-s + 2·29-s − 30-s − 32-s − 6·34-s + 36-s − 2·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.527032931\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.527032931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.00174044127802, −13.37485902304076, −13.12362029465099, −12.34963803899743, −12.00497907912896, −11.47110788984780, −10.66223879767136, −10.40546932185978, −9.940742482827554, −9.455915179702551, −8.768801399068934, −8.553475588057179, −7.953181423599950, −7.454465686836998, −6.896174471983714, −6.243814888809374, −5.881997347324504, −5.147437823532238, −4.523902997032421, −3.677131154087957, −3.276735634043244, −2.592597205963892, −1.863943218062554, −1.393894221128836, −0.5628585020324589,
0.5628585020324589, 1.393894221128836, 1.863943218062554, 2.592597205963892, 3.276735634043244, 3.677131154087957, 4.523902997032421, 5.147437823532238, 5.881997347324504, 6.243814888809374, 6.896174471983714, 7.454465686836998, 7.953181423599950, 8.553475588057179, 8.768801399068934, 9.455915179702551, 9.940742482827554, 10.40546932185978, 10.66223879767136, 11.47110788984780, 12.00497907912896, 12.34963803899743, 13.12362029465099, 13.37485902304076, 14.00174044127802